On a class of two-dimensional incomplete Riemann solvers
This work provides a novel framework for two-dimensional Riemann solvers that can improve accuracy in hyperbolic conservation laws, particularly for magnetohydrodynamics, but the improvements are incremental over existing multidimensional HLL schemes.
The paper proposes a general class of genuinely two-dimensional incomplete Riemann solvers for conservation laws, extending Balsara's multidimensional HLL scheme to PVM/RVM finite volume methods. The methods are applied to magnetohydrodynamics with an efficient divergence cleaning technique, and numerical tests demonstrate good performance on two-dimensional problems.
We propose a general class of genuinely two-dimensional incomplete Riemann solvers for systems of conservation laws. In particular, extensions of Balsara's multidimensional HLL scheme [J. Comput. Phys. 231 (2012) 7476-7503] to two-dimensional PVM/RVM (Polynomial/Rational Viscosity Matrix) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM/RVM flux with two purely two-dimensional PVM/RVM fluxes at vertices, which take into account transversal features of the flow. The proposed methods are applicable to general hyperbolic systems, although in this paper we focus on applications to magnetohydrodynamics. In particular, we propose an efficient technique for divergence cleaning of the magnetic field that provides good results when combined with our two-dimensional solvers. Several numerical tests including genuinely two-dimensional effects are presented to test the performances of the proposed schemes.